Step by step
The question
Question: find the diagonal of a box 3 cm by 4 cm by 5 cm.
The diagonal goes from a bottom corner to the top corner on the other side, right through the box.
1 Floor diagonal
First the floor.
Its diagonal d makes a right triangle with the sides 3 and 4.
Pythagoras: d squared is 3 squared plus 4 squared.
3 times 3 is 9.
4 times 4 is 16.
Together 25.
5 times 5 is 25.
So d is 5.
Careful!
We cannot just add the sides.
A diagonal is a short cut: Pythagoras, twice.
2 Diagonal of the box
Now a new right triangle stands up: the floor diagonal d, the height 5, and the diagonal D of the box.
Pythagoras again: D squared is d squared plus 5 squared.
d squared is 25.
5 times 5 is 25.
Together 50.
50 is not a square number.
A calculator gives 7.071, and so on.
To the nearest tenth, D is about 7.1 cm.
3 Check
Let's check: the two steps are the same as l squared plus w squared plus h squared.
9 plus 16 plus 25 is 50, the same D squared.
So the answer is right!
Let's look back.
First: floor diagonal.
Then: diagonal of the box.
Last: check.